Discussion

In this chapter we presented a first application of the general model Eq. (3.1) for the co-ordinated development of different maps in the visual cortex. We demonstrated that OD segregation can stabilize pinwheels even if they are intrinsically unstable in the uncoupled dynamics of the OP map. We identified and analyzed the stability properties of solutions forming a spatially regular layout with pinwheels arranged in a crystalline array. We calcu-lated phase diagrams showing the stability of these pinwheel crystals in dependence on the OD bias and the inter-map coupling strength.

Our results indicate that the overall dominance of one eye is important for the effectiveness of this mechanism. In this case, OD domains form a system of hexagonal patches rather than stripes enabling the capture and stabilization of pinwheels by inter-map coupling. Supporting this notion, visual cortex around the time of early OP development is indeed dominated by one eye and has a pronounced patchy layout of OD domains [56–58]. Further support for the relevance of this picture comes from experiments in which the OD map was removed artificially resulting in a significantly smoother OP map [33]. We identified three OD so-lutions: stripes, hexagons, and a constant solution, which are stable depending on the OD bias. All three patterns are found to resemble patterns observed in physiological OD maps.

Interestingly, all three types of patterns are found to coexist in the visual cortex of macaque monkeys [56]. For such an OD map our model predicts a systematic variation of the pinwheel density.

To the best of our knowledge we present for the first time an analytical solvable model for the coordinated development of OP and OD maps. We use a dynamical systems approach which allows for a perturbative expansion of the dynamics. Using weakly nonlinear analysis we derived amplitude equations as an approximate description of the dynamics near the pat-tern formation threshold. We identified fixed points and could study their stability properties using different types of inter-map coupling energies. We found that for the low order ver-sion of these energies a strong inter-map coupling can lead to OP map suppresver-sion, causing the orientation selectivity of all neurons to vanish. In contrast, the higher order variants of the coupling energies do not lead to map suppression but only influence pattern selection.

Moreover, we could identify a limit in which the inter-map coupling becomes unidirectional enabling for the use of the uncoupled OD patterns. To confirm our results and to study the impact of a finite backreaction on the OD map we solved the full field dynamics numerically.

In particular, we studied the dynamics of pinwheel crystallization. With the presented ana-lytical approach we were able to show that pinwheel rich solutions correspond to the energetic ground state of the system for large inter-map coupling. This is supported by the fact that pinwheels can actually be created from an initial OP stripe pattern. Pinwheel creation as a result of a pinwheel rich energetic ground state thus serves as a simple test for models of OP development. There is currently only one other model that was shown to be capable of generating pinwheels from pinwheel free initial conditions [49], which will be discussed in the next chapter.

We studied the impact of inter-map coupling on the geometric relationships between OP and OD maps and showed that from observation of map structures one indeed can learn about the optimized energy. From a symmetry-based analysis we identified a class of ener-gies which reflect the experimentally observed geometric relationships between OD and OP maps [68, 73–75]. We thus studied gradient-type as well as product-type coupling energies.

We identified a class of PWC solutions which become stable for large inter-map coupling.

This class depends on a single parameter which is specific to the used inter-map coupling energy. This demonstrates that, although pinwheel stabilization is not restricted to a specific choice of the interaction term, each analyzed phase diagram is specific to the coupling energy.

In the case of the product-type coupling energies the resulting phase diagrams are relatively complex as stationary solutions and stability borders explicitly depend on the OD bias. In contrast, for the gradient-type coupling energies the bias dependence can be absorbed into the coupling strength and only selects the stationary OD pattern. This leads to a rather simple phase diagram. We identified several biologically implausible OP patterns. In the case of the product-type energies we found orientation scotoma solutions which are selective to only two preferred orientations. In the case of the low order gradient-type energy we found OP patterns containing pinwheels with a topological charge of 1 which have not yet been observed in experiments. We further analyzed the stationary patterns with respect to in-tersection angles and pinwheel positions. Remarkably, all analyzed PWC solutions show the tendency for iso-orientation lines to intersect OD borders perpendicularly, even in the case of the product-type energy which is not expected per se. However, for the low order versions of the coupling energy the distribution of intersection angles is not continuous. Half of the pinwheels in the analyzed PWC solutions are located at OD extrema, as expected from the used coupling energies. Some pinwheels, however, are located at OD saddle-points. Remark-ably, such correlations, which are expected from the gradient-type coupling energies, occur also in the case of the product-type energies. However, correlations between pinwheels and OD saddle-points have not yet been studied quantitatively in experiments.

From the analyzed phase diagrams we conclude that the higher order gradient-type coupling energy best reflects the map layouts and their correlations in the visual cortex. As we will see in the next chapter, a clear distinction between the energies with respect to the geometric re-lationships can be made only when considering spatially aperiodic map layouts. We restricted our analysis to four representative examples of coupling energies. The general structure of the amplitude equations is universal and only the coupling coefficients change when changing the coupling energy. We therefore expect that also for other coupling energies, respecting the proposed set of symmetries, PWC solutions become stable for large inter-map coupling.

In this chapter we showed that the presented model offers a solution to the problem of pin-wheel stability. To solve this problem, the influence of OD segregation on OP maps previously had been studied in a series of models such as elastic net models [18, 27–31], self-organizing map models [24, 32–35], spin-like Hamiltonian models [20], spectral filter models [37], corre-lation based models [36], and evolving field models [38]. One should note, however, that the previous models mentioned above are a numerical approach. Whether the obtained solutions

are attractors or just transient states and the solutions further develop towards pinwheel free solutions remains unclear. In particular, it is in principle conceivable that their solutions will converge to crystalline arrangements in long simulations. Moreover, in many previous models a continuous variation of the inter-map coupling strength is not possible which makes it hard to disentangle the contribution of each map.

The presented model questions the widely held view that OD stripes are capable of stabilizing pinwheels. Our analysis shows that OD stripes are indeed not able to stabilize pinwheels, a result that appears to be independent of the specific type of map interaction. Several other theoretical studies, using numerical simulations [18, 38, 99], indicate that a more banded OD pattern leads to a less pinwheel rich OP map.

We presented a thorough characterization of the stable OP and OD patterns. However, the pinwheel crystals we obtained in our model, although beautiful and easy to characterize, in several aspects deviate from experimentally obtained maps. All PWC solutions have a large pinwheel density of about 3.5 or even 5.2 pinwheels per Λ². Densities obtained from experiments, although broadly scattered, are in the range of 3 pinwheels per Λ² [16]. In con-trast to the large variability of map layouts in experimentally observed maps, the pinwheel crystals show a regular and stereotyped structure. The reason for this might be neglected biological constraints in our model such as realistic boundary conditions or biological noise in the system. Another aspect is the developmental stage which is represented by the maps of our model. PWC solutions represent attractor states and thus correspond to late stages of development potentially only comparable to the late adult pattern. One might therefore expect that during the development a further crystallization of the pattern, along with an increase in the pinwheel density takes place. Pinwheel crystals have been previously reported in several abstract [17, 19] as well as feature based models [100]. Remarkably, when modeling receptive field development with synaptic long-range connections the resulting OP map shows a striking similarity to the hexagonal PWC presented in this chapter [101].

We showed that the presented framework can be generalized to include any additional colum-nar system and thus is not restricted to interactions among OD and OP maps. A reason to consider other visual cortical maps originates from the finding that the removal of the OD map in experiments does not completely destabilize pinwheels [33]. Moreover, in tree shrews, animals which lack OD columns, the OP maps contain pinwheels [39]. This might reflect the influence of additional columnar systems like spatial frequency columns that are expected to interact with the OP map in a similar fashion as OD columns [68]. To better account for the non-crystalline layout of experimentally obtained maps we also extended the map interaction model to higher dimensions. In numerical simulations we illustrated inter-map coupling with three and four columnar systems. Although in this case pinwheel stabilization is possible even without an OD bias, the resulting stationary OP patterns are either stripes or PWC solu-tions. Higher feature dimensions are often treated with dimension reduction models [24,102].

A dependence of the number of feature maps on the stabilization of pinwheels was observed for instance in a Kohonen model [100]. We did not consider an interaction with the retino-topic map. Compared to other feature maps, the retinoretino-topic map seems to be exceptional

concerning the geometric relationships. It has been observed that the correlations between OP and retinotopic map are such that high gradient regions do not avoid each other [103].

Such correlations cannot be easily treated with dimension reduction models, see [104]. Re-markably, in our model we identified coupling terms that could account for an attraction of high gradient regions. Such terms contain the gradient of only one field and can thus be considered as a mixture of the gradient and the product-type energy.

Our analysis conclusively demonstrates that OD segregation can stabilize pinwheel crystals even if they are intrinsically unstable in the uncoupled dynamics of the OP map, raising the possibility that inter-map coupling is the mechanism of pinwheel stabilization in the visual cortex. In the next chapter, we study an alternative hypothesis for the origin of pinwheel sta-bilization. Neurons in the visual cortex form extensive connections horizontal to the cortical surface linking different orientation columns [39–48]. It is possible to include these long-range connections in our dynamics by adding non-local interaction terms. It has been shown [49,50]

that with increasing spatial extend of these interactions the layout of maps becomes more irregular.

Inter-Map Coupling

5.1. Introduction

We have seen in the previous chapter that map interactions can lead to a stabilization of pinwheel rich OP patterns. The resulting OP maps, however, are pinwheel crystals and thus spatially periodic. In this chapter we add a source of irregularity by studying a second hypothesis of pinwheel stabilization. It has been demonstrated that the inclusion of long-range interactions in the dynamics leads to stable pinwheel rich OP patterns [49, 50]. Compared to the pinwheel crystals these solutions have a spatially irregular layout. In this chapter we discuss the consequences of long-range interactions in the coupled model.

In particular, we show that the spatially irregular OP layout can be transferred onto the OD map. Similar to the previous chapter we identify a limit in which the backreaction on the dynamics of the OP map can be neglected. We identify an analytically tractable class of stationary solutions and provide their stability criteria. When using identical wavelengths for both maps we show that the OD pattern assumes a patchy layout which resembles the patterns found in cats. For the stable solutions we quantify the geometric relationships showing, for instance, that these relationships are specific to the used type of coupling energy. We further study the impact of the backreaction on the OP map demonstrating that the pinwheel rich, aperiodic solutions are preserved even for a considerable amount of backreaction. We give a potential explanation of the OD layout differences in cats and macaque monkeys. When introducing a detuning in the average wavelength between OP and OD maps as found in macaque monkeys we show that the resulting OD layout is stripe-like rather than patchy, resembling the OD layout found in monkeys.

The coupled dynamics of OD and OP maps we consider in this chapter is given by

∂_tz(x, t) = Lˆ_zz(x, t)−N₃[z, z, z]−δU δz

∂_to(x, t) = Lˆ_oo(x, t)−o(x, t)³−δU

δo . (5.1)

The cubic nonlinearity N3[z, z, z], introduced in Section 3.1, contains local and non-local contributions given by

N₃[z, z, z] = (g−1)|z(x)|²z(x) + 2−g

2πσ² Z

d²y e^{−|x−y|2/2σ2}

z(x)|z(y)|²+1

2z(x)z(y)²

. (5.2)

The dynamics can be derived from the energy functional E[z, o] = −

Z d²x

z(x) ˆL_zz(x) +1

2o(x) ˆL_oo(x) +1

2(1−g)|z(x)|⁴−1 4o(x)⁴

+2−g 4πσ²

Z d²x

Z d²y

|z(x)|²|z(y)|²+ 1

2z((y))²z²(x)

e^{−|x−y|2/(2σ2)} +

Z

d²x U . (5.3)

Non-local interactions are introduced only for the OP part of the dynamics. Experiments suggest that long-range horizontal connections are sensitive to the preferred orientation of the neurons which are interconnected while their correlations to OD is less clear, see Section 2.3.

Compared to the previous chapter the dynamics is invariant under the inversion symmetry o(x)→ −o(x) and the uncoupled OD attractors are thus stripes forr_o>0. As in the previous chapter we study the four representative examples of coupling energiesU given in Eq. (3.23).

5.2. Permutation symmetry

When we rewrite the nonlinearity Eq. (5.2) in the following form N[u, v, w] = (g−1)u(x)v(x)w(x)

+2−g 4πσ²

Z

d²y e^{−|x−y|2/(2σ2)}

u(x)v(y)w(y) + v(x)w(y)u(y) +w(x)u(y)v(y)

, (5.4)

we see that the nonlinearity is invariant under a cyclic permutation of the fields i.e.

N[u, v, w] =N[v, w, u]. (5.5)

This permutation symmetry has far reaching consequences on the attractor states of the dynamics. In the following we give a brief review of the uncoupled OP dynamics with permu-tation symmetry. Consequences of the presence or absence of this symmetry are also discussed in Chapter 6 and 7.

5.3. The uncoupled dynamics

Without inter-map coupling the dynamics for the OP map is studied in [49,50]. In this section we briefly review the main results of this work. In case of the uncoupled dynamics we can use the low order amplitude equations derived in Section 4.3. With the inclusion of the non-local

interaction terms Eq. (5.2) the coupling coefficients, see Eq. (4.59), are given by g_ij = g+ (2−g)

e^−12σ2(^~ki−~k_j)² +e^−12σ2(^~ki+~k_j)² g_ii = 1 + 1

2(2−g)e^−2σ2 f_ij = g+ (2−g)

e^−12σ2(^~ki−~kj)² +e^−12σ2(^~ki+~kj)²

f_ii = 0. (5.6)

If the directions of the wave vectors ~k_j = (cosα_j,sinα_j)k_c,z are represented by angles α_j then the coefficients g_ij and f_ij are functions only of the angle α = |α_i −α_j| between the wavevectors ~ki and~kj. The coefficients can be obtained from the continuous functions

g(α) = e^−ı~k0~x

N3[e^ı~k0~x, e^ıh(α)~x, e^−ıh(α)~x] +N3[e^ıh(α)~x, e^ı~k0~x, e^−ıh(α)~x] f(α) = e^−ı~k0~x

N₃[e^ıh(α)~x, e^−ıh(α)~x, e^ı~k0~x] +N₃[e^−ıh(α)~x, e^ıh(α)~x, e^ı~k0~x]

, (5.7) with ~k0 = (1,0)k_c,z and h(α) = (cosα,sinα)k_c,z. The coefficients are obtained as g_ij = g(|α_i −α_j|), g_ii = g(0)/2, g_ij− = g(|α_i −α_j +π|) and f_ij = f(|α_i−α_j|). In case of the coupling coefficients Eq. (5.6) the coupling function is given by

g(α) = g+ (2−g)

e^{−σ2(1+cosα)}+e^{−σ2(1−cosα)} f(α) = g+ (2−g)

e^{−σ2(1−1 cosα)}+e^{−σ2(1+cosα)}

. (5.8)

Whereas f(α) is π-periodic, g(α) is 2π-periodic in general. The permutation symmetry Eq. (5.5), however, impliesg(α+π) =g(α).

Close to the bifurcation point a large class of attractors to the uncoupled OP dynamics is given by

z(x) =

nz

X

j

Aje^ıφje^ılj~kj~x. (5.9) The binary variable l_j = ±1 determines whether the mode with wavevector ~kj or −~kj is active. For this class of solutions opposite modes are suppressed i.e. A_j− = 0. Planforms of this type are therefore called Essentially Complex Planforms (ECP). For an ECP the field z(x) cannot become real and thus the OP map is selective to all possible stimulus orientations.

The phases of the active modes φj are arbitrary and independent of the mode configuration l_j. Due to permutation symmetry g_ij is a circulant matrix g_ij =g_{(j−i) modn}. Since P

jg_ij is in this case independent of the index ithe stationary solutions are uniform and given by

Ai =A=

s r_z P

jg_ij . (5.10)

Figure 5.1.: Phase diagram of the uncoupled OP dynamics. (a) Essentially complex planforms with different numbersn_z =n= 1,2,3,5,15 of active modes. The diagrams to the left of each pattern display the position of the wavevectors of active modes on the critical circle. For n= 3, there are two patterns; for n= 5, there are four, and forn= 15, there are 612 different patterns. (b) Phase diagram of the uncoupled case. Shown are the regions where a essential complex planform with nmodes has the minimum energy. If non-local interactions are dominant (g <1) and long-ranging (σ large compared to Λ), quasiperiodic planforms are selected. Reproduced from [49].

For a given n_z there are 2^nz possible ECP configurations, however, many of which can be transferred into each other by a rotation or reflection thus defining equivalence classes. The actual number of distinct classes is smaller but nevertheless grows exponentially with n_z. Their phase diagram is shown for the two model parameters g and σ/Λ in Fig. 5.1(b). For g > 1 only n_z = 1 solutions are energetically preferred while for g < 1 and σ/Λ large enough solutions with more modes n_z > 1 are energetically preferred. The different ECP solutions substantially vary in their pinwheel density. In the following we denote LDP as the low pinwheel density planform given byl_j = 1,∀j and HDP as thehigh pinwheel density planform given by l_j = 1 forjeven andl_j =−1 for j odd. Note, the HDP configuration may not be the one with the maximum pinwheel density but has a pinwheel density larger than ρ=π. How the pinwheel density varies with the different planforms is detailed in Chapter 6.

The model has a vast number of multistable solutions. For a given n_z there is a multistability of different ECP solutions, see Fig. 5.1(a). As shown in [49,50] permutation symmetry ensures that for a givenn_z all ECP solutions share the same energy and stability properties. Realistic patterns are obtained for g < 1 and σ ≫ Λ, see Fig. 5.1(b). In the case of n_z ≥ 4 the corresponding OP map is a spatially quasiperiodic pattern becoming more and more irregular with increasing n_z. For intermediate and large n_z these patterns resemble those obtained in experiments.

Without inter-map coupling the OD maps have a regular stripe layout given by

This layout does not resemble the patchy layout observed in cats. Even the more stripe-like layouts found in monkeys and humans where OD bands meander, occasionally branching and terminating, are much more irregular than OD stripes.

In the following we study how inter-map coupling influences the layout of OD and OP maps.

In particular, we are interested in a limit in which we can neglect the backreaction on the OP map. Besides a substantial simplification this limit allows us to use the uncoupled OP attractors which, for large n_z, resembles the OP layout found in physiological maps. We mainly concentrate on planforms with n_z ≥ 4 because in contrast to the last chapter these solutions have a spatially irregular layout.

5.4. Low order coupling energies

In this section we study the impact of inter-map coupling on the layout of OD maps using the low order inter-map coupling energy terms in Eq. (3.23). We show that, similar to the case of PWC solutions, there is a suppression of OD leading to a completely unselective OD map. Close to the bifurcation point r_z = 0, r_o = 0 stationary solutions to the dynamics Eq. (5.1) are calculated using weakly nonlinear analysis. The general derivation of coupled amplitude equations up to seventh order is given in Section 4.3. In contrast to the previous

In this section we study the impact of inter-map coupling on the layout of OD maps using the low order inter-map coupling energy terms in Eq. (3.23). We show that, similar to the case of PWC solutions, there is a suppression of OD leading to a completely unselective OD map. Close to the bifurcation point r_z = 0, r_o = 0 stationary solutions to the dynamics Eq. (5.1) are calculated using weakly nonlinear analysis. The general derivation of coupled amplitude equations up to seventh order is given in Section 4.3. In contrast to the previous

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